2. Remember
You just invented a “magic math pill” that will increase test scores.
On the day of the first test you give the pill to 4 subjects. When these same subjects take the second test they do not get a pill
Did the pill increase their test scores?

3. What if. . .
You just invented a “magic math pill” that will increase test scores.
On the day of the first test you give a full pill to 4 subjects. When these same subjects take the second test they get a placebo. When these same subjects that the third test they get no pill.

4. Note You have more than 2 groups You have a repeated measures design
You need to conduct a Repeated Measures ANOVA

5. Tests to Compare Means Design of experiment
Independent Variables and # of levels
Independent Samples
Related Samples
One IV, 2 levels
Independent t-test
Paired-samples t-test
One IV, 2 or more levels
ANOVA
Repeated measures ANOVA
Tow IVs, each with 2 or more levels
Factorial ANOVA
Repeated measures factorial ANOVA

6. What if. . .
You just invented a “magic math pill” that will increase test scores.
On the day of the first test you give a full pill to 4 subjects. When these same subjects take the second test they get a placebo. When these same subjects that the third test they get no pill.

7. Results Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 7678
Sub 4 93 92 96
Mean

8. For now . . . Ignore that it is a repeated design
Pill
Placebo
No Pill
Sub 1 57 60 64
Sub 2 71 72 74
Sub 3 75 76 78
Sub 4 93 92 96
Mean

9. Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 7893 92 96
Mean
Between Variability = low

10. Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 7893 92 96
Mean
Within Variability = high

11. Source df SS MS F
Drug 2
34.67
17.33
.09
Within 9
1720
191.11
Total 11

12. Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78
Notice – the within variability of a group can be predicted by the other groups
Pill
Placebo
No Pill
Sub 1 57 60 64
Sub 2 71 72 74
Sub 3 75 76 78
Sub 4 93 92 96
Mean

13. Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78
Notice – the within variability of a group can be predicted by the other groups
Pill
Placebo
No Pill
Sub 1 57 60 64
Sub 2 71 72 74
Sub 3 75 76 78
Sub 4 93 92 96
Mean
Pill and Placebo r = .99; Pill and No Pill r = .99; Placebo and No Pill r = .99

14. Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.3375 76 78
76.33
Sub 4 93 92 96
93.66
These scores are correlated because, in general, some subjects tend to do very well and others tended to do very poorly

15. Repeated ANOVA
Some of the variability of the scores within a group occurs due to the mean differences between subjects.
Want to calculate and then discard the variability that comes from the differences between the subjects.

16. Example Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74
72.33
Sub 3 75 76 78
76.33
Sub 4 93 92 96
93.66
75.66

17. Sum of Squares SS Total Computed the same way as before
The total deviation in the observed scores
Computed the same way as before

18. Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.3375 76 78
76.33
Sub 4 93 92 96
93.66
75.66
SStotal = ( )2+ ( ) ( )2 =
*What makes this value get larger?

19. Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.3375 76 78
76.33
Sub 4 93 92 96
93.66
75.66
SStotal = ( )2+ ( ) ( )2 =
*What makes this value get larger?
*The variability of the scores!

20. Sum of Squares SS Subjects
Represents the SS deviations of the subject means around the grand mean
Its multiplied by k to give an estimate of the population variance (Central limit theorem)

21. Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.3375 76 78
76.33
Sub 4 93 92 96
93.66
75.66
SSSubjects = 3(( )2+ ( ) ( )2) = 1712
*What makes this value get larger?

22. Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.3375 76 78
76.33
Sub 4 93 92 96
93.66
75.66
SSSubjects = 3(( )2+ ( ) ( )2) = 1712
*What makes this value get larger?
*Differences between subjects

23. Sum of Squares SS Treatment
Represents the SS deviations of the treatment means around the grand mean
Its multiplied by n to give an estimate of the population variance (Central limit theorem)

24. Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.3375 76 78
76.33
Sub 4 93 92 96
93.66
75.66
SSTreatment = 4(( )2+ ( )2+( )2) = 34.66
*What makes this value get larger?

25. Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.3375 76 78
76.33
Sub 4 93 92 96
93.66
75.66
SSTreatment = 4(( )2+ ( )2+( )2) = 34.66
*What makes this value get larger?
*Differences between treatment groups

26. Sum of Squares Have a measure of how much all scores differ
SSTotal
Have a measure of how much this difference is due to subjects
SSSubjects
Have a measure of how much this difference is due to the treatment condition
SSTreatment
To compute error simply subtract!

27. Sum of Squares SSError = SSTotal - SSSubjects – SSTreatment
8.0 =

28. Source df SS
Subjects
Treatment
34.66
Error
8.00
Total 11

29. Source df SS MS F
Drug 2
34.67
17.33
.09
Within 9
1720
191.11
Total 11

30. Compute df Source df SS Subjects 1712.00 Treatment 34.66 Error 8.00
df total = N -1
Source df SS
Subjects
Treatment
34.66
Error
8.00
Total 11

31. Compute df Source df SS Subjects 3 1712.00 Treatment 34.66 Error 8.00
df total = N -1
df subjects = n – 1
Source df SS
Subjects 3
Treatment
34.66
Error
8.00
Total 11

32. Compute df Source df SS Subjects 3 1712.00 Treatment 2 34.66 Error
df total = N -1
df subjects = n – 1
df treatment = k-1
Source df SS
Subjects 3
Treatment 2
34.66
Error
8.00
Total 11

33. Compute df Source df SS Subjects 3 1712.00 Treatment 2 34.66 Error 6
df total = N -1
df subjects = n – 1
df treatment = k-1
df error = (n-1)(k-1)
Source df SS
Subjects 3
Treatment 2
34.66
Error 6
8.00
Total 11

34. Compute MS Source df SS MS Subjects 3 1712.00 Treatment 2 34.66 17.33
Error 6
8.00
Total 11

35. Compute MS Source df SS MS Subjects 3 1712.00 Treatment 2 34.66 17.33
Error 6
8.00
1.33
Total 11

36. Compute F Source df SS MS F Subjects 3 1712.00 Treatment 2 34.66 17.33
13.00
Error 6
8.00
1.33
Total 11

37. Test F for Significance
Source df SS MS F
Subjects 3
Treatment 2
34.66
17.33
13.00
Error 6
8.00
1.33
Total 11

38. Test F for Significance
Source df SS MS F
Subjects 3
Treatment 2
34.66
17.33
13.00*
Error 6
8.00
1.33
Total 11
F(2,6) critical = 5.14

39. Practice
You wonder if the statistic tests are of all equal difficulty. To investigate this you examine the scores 4 students got on the three different tests

40. Test 1
Test 2
Test 3
Sub 1 60 70 78
Sub 2 76 85
Sub 3 64 90 89
Sub 4 77 81 94

41. Source df SS MS F
Subjects
Treatment
564.50
Error
234.83
Total

43. Source df SS MS F
Subjects 3
366.17
Treatment 2
564.50
282.25
7.21*
Error 6
234.83
39.13
Total 11

46. Practice
Sleep researchers decide to test the impact of REM sleep deprivation on a computerized assembly line task. Subjects are required to participate in two nights of testing. On each night of testing the subject is allowed a total of four hours of sleep. However, on one of the nights, the subject is awakened immediately upon achieving REM sleep. Subjects then took a cognitive test which assessed errors in judgment. Did sleep deprivation effect subjects cognitive ability?

47. REM Deprived
Control Condition 26 20 15 4 8 9 44 36

48. tobs = 3.04*
Sleep deprivation effected their cognitive abilities.

49. No Class on Monday!